To simplify the expressions provided, let's go through each of them one by one.
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Simplifying \( \sqrt[3]{108c^{17}} \):
We can simplify this expression by breaking it down: \[ \sqrt[3]{108c^{17}} = \sqrt[3]{108} \cdot \sqrt[3]{c^{17}} \] The cube root of \(108\) can be simplified: \[ 108 = 27 \times 4 = 3^3 \times 4 \Rightarrow \sqrt[3]{108} = 3\sqrt[3]{4} \] For \(c^{17}\): \[ \sqrt[3]{c^{17}} = c^{\frac{17}{3}} = c^5 \cdot c^{\frac{2}{3}} \] Therefore: \[ \sqrt[3]{108c^{17}} = 3\sqrt[3]{4} \cdot c^5 \cdot c^{\frac{2}{3}} = 3c^5 \sqrt[3]{4c^2} \]
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Simplifying \( 3c^5 \sqrt[3]{4c^2} \):
This is already a simplified expression, so we keep it as: \[ 3c^5 \sqrt[3]{4c^2} \]
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Simplifying \( 3 \sqrt[3]{4c^{17}} \):
Likewise, we can express this as: \[ 3 \sqrt[3]{4c^{17}} = 3 \cdot \sqrt[3]{4} \cdot \sqrt[3]{c^{17}} = 3 \cdot \sqrt[3]{4} \cdot c^5 \cdot c^{\frac{2}{3}} = 3c^5 \sqrt[3]{4c^2} \]
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Simplifying \( 18c^8 \sqrt{3c} \):
We can break it down: \[ \sqrt{3c} = \sqrt{3} \cdot \sqrt{c} \] So, we rewrite it as: \[ 18c^8 \sqrt{3c} = 18c^8 \cdot \sqrt{3} \cdot \sqrt{c} = 18c^{8.5}\sqrt{3} \]
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Simplifying \( 6c^8 \sqrt{3c} \):
Similarly: \[ 6c^8 \sqrt{3c} = 6c^8 \cdot \sqrt{3} \cdot \sqrt{c} = 6c^{8.5}\sqrt{3} \]
Summary of Simplified Results:
- \( 3c^5 \sqrt[3]{4c^2} \)
- \( 3c^5 \sqrt[3]{4c^2} \)
- \( 3c^5 \sqrt[3]{4c^2} \)
- \( 18c^{8.5}\sqrt{3} \)
- \( 6c^{8.5}\sqrt{3} \)
As you can see, the first three expressions evaluate to the same form (\( 3c^5 \sqrt[3]{4c^2} \)). The last two expressions have different coefficients but share the same variable structure with respect to \(c\).
If needed, you can combine \(18c^{8.5}\sqrt{3} \) and \(6c^{8.5}\sqrt{3} \) together if representing them in a simplified form is intended.